Tipler: 21-7 Electromagnetism I Electric Dipoles and their Interactions in Electric Fields

An electric dipole consists of a positive charge separated from a negative charge of the same magnitude by a small distance. Which, if any, of the diagrams best represents the electric field lines around an electric dipole? ET EP

Learning Objectives To calculate E produced by an electric dipole To investigate the forces and torques an electric dipole experiences in an external uniform E-field

An electric dipole: l +q+q-q-q Electrically neutral A Reminder of Electric Dipoles Define dipole moment as p = q  l The vector of p is drawn from the negative to the positive point charge p is a vector. p

Molecular example: H 2 O p= 6.1 x C m The electric dipole of water makes it an excellent solvent Used in heating food in a microwave oven

Calculation of the E-field at an arbitrary (r,  )

The E-field due to an Electric Dipole - Calculation To simplify the calculation, we will only compute the field along the axis r E -q+q a a/2

For r >> a This applies to any points along the line of the dipole, and only for points along the line of the dipole. Along this particular direction, the E field from the positive charge is in opposite direction to that from the negative charge.

To obtain E-field: 1)Coulomb’s Law, involving vector sums. 2)Gauss’s law, if the charge distribution has a high degree of symmetry. 3) Get V first, then differentiate. No vector sums. How to make a decision: Choose the method that consumes the least number of brain cells.

l +q+q-q-q V due to the two charges at P Assumption: r >> l r-r- r+r+ P  r l/2 No nasty vectors here.

Hence as r >> l The sign of V p depends on .

In Plane Polar coordinates (lecture 4)

l +q+q l/2 r-r- r+r+ P  r -q-q E r points towards the centre of the dipole E T points towards decreasing . E r change sign when  is greater than 90 degrees.

Electric Dipoles in Uniform Electric Fields (Tipler )

-q-q +q+q p l E qEqE -qE No Net Force But Torque  - rotates the dipole clockwise Two equal and opposite forces whose lines of action do not coincide constitute a couple. The two forces always have a turning effect, called a torque

A torque is defined as the moment of a force Mathematically (STMR)

Torque (of a couple) (Tipler – pages ) The resultant torque is: The magnitude of the torque of a couple is calculated from i.e. torque = one force  perpendicular distance between forces

-q-q +q+q l qE -qE Torque   d The torque tends to align p and E

In vector form: The direction of  p E 

An electric dipole of moment p is placed in a uniform external electric field. The dipole moment vector is in the positive y direction. The external electric field vector is in the positive x direction. When the dipole is aligned as shown in the diagram, the net torque is in the A)positive x direction. B)positive y direction. C)negative x direction. D)positive z direction. E)negative z direction.

Electric potential energy of a dipole

+q+q -q  E Electric potential at +q is V +, the potential energy is qV + Electric potential at -q is V -, the potential energy is -qV - The total potential energy of the dipole is:

Thus the P.E. of an electric dipole in an E-field is: Minimum at  = 0, maximum at  = , and zero at  =  /2

The electric dipole is like an electric version of a compass P E The potential energy of a magnetic dipole is What is the torque on the dipole for the above configuration?

Cooking instructions: Molecules with dipole moment, Molecules that are mobile H 2 O= 6.1 x C m

Electric Dipole Moment of Some Gas Molecules HCl 1.1 HBr 0.8 H 2 O1.8 SO N 2 O0.2 NH 3 1.5

An electric dipole in an electric field experiences a torque: The potential energy for an electric dipole in an electric field E depends on the orientation of the dipole moment p with respect to the field:

p = 0.02 e.nm E = 3 × 10 3 N/C Calculate (a)the magnitude of the torque (b)The potential energy

Review and Summary An electric dipole is a pair of electric charges of equal magnitude q but opposite sign, separated by a distance l The electric dipole moment is defined to have magnitude p = ql We calculated the E of an electric dipole at any position in space by a method far easier than using Coulomb’s law and superposition